Geom::Affine Class Reference

3x3 matrix representing an affine transformation. More...

#include <affine.h>

Inheritance diagram for Geom::Affine:

Public Member Functions
	Affine ()
	Affine (Coord c0, Coord c1, Coord c2, Coord c3, Coord c4, Coord c5)
	Create a matrix from its coefficient values. More...
Coord	operator[] (unsigned i) const
	Access a coefficient by its index. More...
Coord &	operator[] (unsigned i)
bool	operator== (Affine const &o) const
Combine with other transformations
Affine &	operator*= (Affine const &m)
	Combine this transformation with another one. More...
Affine &	operator*= (Translate const &t)
	Combine this transformation with a translation. More...
Affine &	operator*= (Scale const &s)
	Combine this transformation with scaling. More...
Affine &	operator*= (Rotate const &r)
	Combine this transformation a rotation. More...
Affine &	operator*= (HShear const &h)
	Combine this transformation with horizontal shearing (skew). More...
Affine &	operator*= (VShear const &v)
	Combine this transformation with vertical shearing (skew). More...
Affine &	operator*= (Zoom const &)
Get the parameters of the matrix's transform
Point	xAxis () const
Point	yAxis () const
Point	translation () const
	Gets the translation imparted by the Affine. More...
Coord	expansionX () const
	Calculates the amount of x-scaling imparted by the Affine. More...
Coord	expansionY () const
	Calculates the amount of y-scaling imparted by the Affine. More...
Point	expansion () const
Modify the matrix
void	setXAxis (Point const &vec)
void	setYAxis (Point const &vec)
void	setTranslation (Point const &loc)
	Sets the translation imparted by the Affine. More...
void	setExpansionX (Coord val)
void	setExpansionY (Coord val)
void	setIdentity ()
	Sets this matrix to be the Identity Affine. More...
Inspect the matrix's transform
bool	isIdentity (Coord eps=EPSILON) const
	Check whether this matrix is an identity matrix. More...
bool	isTranslation (Coord eps=EPSILON) const
	Check whether this matrix represents a pure translation. More...
bool	isScale (Coord eps=EPSILON) const
	Check whether this matrix represents pure scaling. More...
bool	isUniformScale (Coord eps=EPSILON) const
	Check whether this matrix represents pure uniform scaling. More...
bool	isRotation (Coord eps=EPSILON) const
	Check whether this matrix represents a pure rotation. More...
bool	isHShear (Coord eps=EPSILON) const
	Check whether this matrix represents pure horizontal shearing. More...
bool	isVShear (Coord eps=EPSILON) const
	Check whether this matrix represents pure vertical shearing. More...
bool	isNonzeroTranslation (Coord eps=EPSILON) const
	Check whether this matrix represents a pure nonzero translation. More...
bool	isNonzeroScale (Coord eps=EPSILON) const
	Check whether this matrix represents pure, nonzero scaling. More...
bool	isNonzeroUniformScale (Coord eps=EPSILON) const
	Check whether this matrix represents pure, nonzero uniform scaling. More...
bool	isNonzeroRotation (Coord eps=EPSILON) const
	Check whether this matrix represents a pure, nonzero rotation. More...
bool	isNonzeroNonpureRotation (Coord eps=EPSILON) const
	Check whether this matrix represents a non-zero rotation about any point. More...
Point	rotationCenter () const
	For a (possibly non-pure) non-zero-rotation matrix, calculate the rotation center. More...
bool	isNonzeroHShear (Coord eps=EPSILON) const
	Check whether this matrix represents pure, nonzero horizontal shearing. More...
bool	isNonzeroVShear (Coord eps=EPSILON) const
	Check whether this matrix represents pure, nonzero vertical shearing. More...
bool	isZoom (Coord eps=EPSILON) const
	Check whether this matrix represents zooming. More...
bool	preservesArea (Coord eps=EPSILON) const
	Check whether the transformation preserves areas of polygons. More...
bool	preservesAngles (Coord eps=EPSILON) const
	Check whether the transformation preserves angles between lines. More...
bool	preservesDistances (Coord eps=EPSILON) const
	Check whether the transformation preserves distances between points. More...
bool	flips () const
	Check whether this transformation flips objects. More...
bool	isSingular (Coord eps=EPSILON) const
	Check whether this matrix is singular. More...
Compute other matrices
Affine	withoutTranslation () const
Affine	inverse () const
	Compute the inverse matrix. More...
Compute scalar values
Coord	det () const
	Calculate the determinant. More...
Coord	descrim2 () const
	Calculate the square of the descriminant. More...
Coord	descrim () const
	Calculate the descriminant. More...

Static Public Member Functions
static Affine	identity ()

Private Attributes
Coord	_c [6]

Related Functions
(Note that these are not member functions.)
std::ostream &	operator<< (std::ostream &out_file, const Geom::Affine &m)
	Print out the Affine (for debugging). More...
Affine	identity ()
	Create an identity matrix. More...
Affine	elliptic_quadratic_form (Affine const &m)
	Given a matrix m such that unit_circle = m*x, this returns the quadratic form x*A*x = 1. More...
bool	are_near (Affine const &a, Affine const &b, Coord eps)
	Nearness predicate for affine transforms. More...

Detailed Description

3x3 matrix representing an affine transformation.

Affine transformations on elements of a vector space are transformations which can be expressed in terms of matrix multiplication followed by addition ( \(x \mapsto A x + b\)). They can be thought of as generalizations of linear functions ( \(y = a x + b\)) to vector spaces. Affine transformations of points on a 2D plane preserve the following properties:

• Colinearity: if three points lie on the same line, they will still be colinear after an affine transformation.
• Ratios of distances between points on the same line are preserved
• Parallel lines remain parallel.

All affine transformations on 2D points can be written as combinations of scaling, rotation, shearing and translation. They can be represented as a combination of a vector and a 2x2 matrix, but this form is inconvenient to work with. A better solution is to represent all affine transformations on the 2D plane as 3x3 matrices, where the last column has fixed values.

\[ A = \left[ \begin{array}{ccc} c_0 & c_1 & 0 \\ c_2 & c_3 & 0 \\ c_4 & c_5 & 1 \end{array} \right]\]

We then interpret points as row vectors of the form \([p_X, p_Y, 1]\). Applying a transformation to a point can be written as right-multiplication by a 3x3 matrix ( \(p' = pA\)). This subset of matrices is closed under multiplication - combination of any two transforms can be expressed as the multiplication of their matrices. In this representation, the \(c_4\) and \(c_5\) coefficients represent the translation component of the transformation.

Matrices can be multiplied by other more specific transformations. When multiplying, the transformations are applied from left to right, so for example m = a * b means: m first transforms by a, then by b.

Definition at line 60 of file affine.h.

Constructor & Destructor Documentation

Affine() [1/2]

Geom::Affine::Affine ( )

inline

Definition at line 73 of file affine.h.

References _c.

Referenced by operator*=().

Affine() [2/2]

Geom::Affine::Affine ( Coord  c0, Coord  c1, Coord  c2, Coord  c3, Coord  c4, Coord  c5  )

inline

Create a matrix from its coefficient values.

It's rather inconvenient to directly create matrices in this way. Use transform classes if your transformation has a geometric interpretation.

See also

Translate

Scale

Rotate

HShear

VShear

Zoom

Definition at line 87 of file affine.h.

References _c.

Member Function Documentation

descrim()

Coord Geom::Affine::descrim

(

)

const

Calculate the descriminant.

If the matrix doesn't contain a shearing or non-uniform scaling component, this value says how will the length of any line segment change after applying this transformation to arbitrary objects on a plane. The new length will be

line_seg.length() * m.descrim()) 

Returns

\(\sqrt{|\det A|}\).

Definition at line 434 of file affine.cpp.

References descrim2(), and Geom::sqrt().

Referenced by Inkscape::UI::PathManipulator::_getStrokeTolerance(), Inkscape::UI::Tools::PencilTool::_interpolate(), Inkscape::UI::Tools::PencilTool::_sketchInterpolate(), _ungroup_compensate_source_transform(), Inkscape::CanvasItemGridAxonom::_update(), Inkscape::DrawingGlyphs::_updateItem(), Inkscape::UI::Tools::PencilTool::addPowerStrokePencil(), Inkscape::UI::Dialog::CloneTiler::apply(), Inkscape::UI::Widget::calcScaleLineWidth(), Inkscape::CanvasItemBpath::closest_distance_to(), SPText::description(), Inkscape::LivePathEffect::LPEOffset::doBeforeEffect(), Inkscape::LivePathEffect::LPESimplify::doEffect(), SPItem::doWriteTransform(), Inkscape::UI::Tools::SpiralTool::finishItem(), Inkscape::UI::Tools::StarTool::finishItem(), Inkscape::CanvasItemRect::get_shadow_size(), SPDesktop::DesktopAffine::getZoom(), item_find_paths(), Inkscape::UI::Toolbar::TextToolbar::lineheight_unit_changed(), Inkscape::LivePathEffect::lpe_shape_revert_stroke_and_fill(), modify_filter_gaussian_blur_from_item(), new_filter_simple_from_item(), objects_query_blur(), objects_query_fontnumbers(), objects_query_strokewidth(), Geom::Ellipse::operator*=(), Inkscape::LivePathEffect::ScalarParam::param_transform_multiply(), Inkscape::LivePathEffect::PowerStrokePointArrayParam::param_transform_multiply(), path_simplify(), SPPath::removeTransformsRecursively(), Inkscape::Filters::FilterDiffuseLighting::render_cairo(), Inkscape::Filters::FilterSpecularLighting::render_cairo(), SPOffset::set_shape(), SPLine::set_transform(), SPPath::set_transform(), SPText::set_transform(), SPGenericEllipse::set_transform(), SPFlowtext::set_transform(), Inkscape::UI::Widget::StrokeStyle::setStrokeDash(), sp_desktop_apply_css_recursive(), sp_gradient_convert_to_userspace(), sp_selected_path_create_offset_object(), sp_selected_path_do_offset(), sp_svg_transform_write(), sp_te_adjust_kerning_screen(), sp_te_adjust_linespacing_screen(), sp_te_adjust_rotation_screen(), sp_te_apply_style(), Inkscape::UI::Tools::sp_tweak_color_recursive(), Inkscape::UI::Tools::sp_tweak_dilate_recursive(), Inkscape::Extension::Internal::PrintLatex::stroke(), stroke_average_width(), take_style_from_item(), text_flow_into_shape(), Inkscape::UI::Toolbar::TextToolbar::text_outer_set_style(), text_put_on_path(), Inkscape::DrawingItem::update(), SPHatchPath::update(), SPShape::update(), and wrap_transforms().

descrim2()

Coord Geom::Affine::descrim2

(

)

const

Calculate the square of the descriminant.

This is simply the absolute value of the determinant.

Returns

\(|\det A|\).

Definition at line 424 of file affine.cpp.

References det().

Referenced by Geom::Parallelogram::area(), descrim(), PatternKnotHolderEntityScale::knot_grabbed(), preservesArea(), and wrap_transforms().

det()

Coord Geom::Affine::det

(

)

const

Calculate the determinant.

Returns

\(\det A\).

Definition at line 416 of file affine.cpp.

References _c.

Referenced by Inkscape::DrawingPattern::_updateItem(), Geom::decompose_degenerate(), descrim2(), flips(), inverse(), isSingular(), max_expansion(), Inkscape::UI::Widget::Stores::snapshot_combine(), and wrap_transforms().

expansion()

Point Geom::Affine::expansion ( ) const

inline

Definition at line 123 of file affine.h.

References expansionX(), and expansionY().

expansionX()

double Geom::Affine::expansionX

(

)

const

Calculates the amount of x-scaling imparted by the Affine.

This is the scaling applied to the original x-axis region. It is \emph{not} the overall x-scaling of the transformation. Equivalent to L2(m.xAxis()).

Definition at line 64 of file affine.cpp.

References _c, and Geom::sqrt().

Referenced by Inkscape::Extension::Internal::SvgBuilder::_flushText(), Inkscape::Filters::FilterGaussian::area_enlarge(), Inkscape::Filters::FilterMorphology::area_enlarge(), Inkscape::Filters::FilterGaussian::complexity(), Inkscape::Filters::FilterMorphology::complexity(), Inkscape::LivePathEffect::LPEMeasureSegments::doBeforeEffect(), expansion(), Geom::Parallelogram::maxExtent(), Geom::Parallelogram::minExtent(), Inkscape::LivePathEffect::NodeSatelliteArrayParam::param_transform_multiply(), Inkscape::Filters::FilterDisplacementMap::render_cairo(), Inkscape::Filters::FilterGaussian::render_cairo(), Inkscape::Filters::FilterMorphology::render_cairo(), Inkscape::Filters::FilterOffset::render_cairo(), setExpansionX(), Inkscape::UI::Tools::MeasureTool::setMeasureItem(), Inkscape::Extension::Internal::LaTeXTextRenderer::sp_flowtext_render(), Inkscape::UI::Tools::spdc_check_for_and_apply_waiting_LPE(), Inkscape::Extension::Internal::PrintEmf::text(), Inkscape::Extension::Internal::PrintWmf::text(), and wrap_transforms().

expansionY()

double Geom::Affine::expansionY

(

)

const

Calculates the amount of y-scaling imparted by the Affine.

This is the scaling applied before the other transformations. It is \emph{not} the overall y-scaling of the transformation. Equivalent to L2(m.yAxis()).

Definition at line 71 of file affine.cpp.

References _c, and Geom::sqrt().

Referenced by Inkscape::Filters::FilterGaussian::area_enlarge(), Inkscape::Filters::FilterMorphology::area_enlarge(), Inkscape::Filters::FilterGaussian::complexity(), Inkscape::Filters::FilterMorphology::complexity(), Inkscape::LivePathEffect::LPEMeasureSegments::doBeforeEffect(), expansion(), Geom::Parallelogram::maxExtent(), Geom::Parallelogram::minExtent(), Inkscape::LivePathEffect::NodeSatelliteArrayParam::param_transform_multiply(), Inkscape::Filters::FilterDisplacementMap::render_cairo(), Inkscape::Filters::FilterGaussian::render_cairo(), Inkscape::Filters::FilterMorphology::render_cairo(), Inkscape::Filters::FilterOffset::render_cairo(), setExpansionY(), Inkscape::UI::Tools::spdc_check_for_and_apply_waiting_LPE(), Inkscape::Extension::Internal::PrintEmf::text(), Inkscape::Extension::Internal::PrintWmf::text(), and wrap_transforms().

flips()

bool Geom::Affine::flips

(

)

const

Check whether this transformation flips objects.

A transformation flips objects if it has a negative scaling component.

Definition at line 368 of file affine.cpp.

References det().

Referenced by Geom::TEST().

identity()

static Affine Geom::Affine::identity ( )

inlinestatic

Referenced by Geom::identity(), Geom::TEST(), and Geom::OptRect::transformTo().

inverse()

Affine Geom::Affine::inverse

(

)

const

Compute the inverse matrix.

Inverse is a matrix (denoted \(A^{-1}\)) such that \(AA^{-1} = A^{-1}A = I\). Singular matrices have no inverse (for example a matrix that has two of its columns equal). For such matrices, the identity matrix will be returned instead.

Parameters

eps	Numerical tolerance

Returns

Inverse of the matrix, or the identity matrix if the inverse is undefined.

Postcondition

(m * m.inverse()).isIdentity() == true

Definition at line 388 of file affine.cpp.

References _c, det(), Geom::max(), Geom::rel_error_bound(), setIdentity(), and Geom::sqrt().

Referenced by SPHatch::_calculateStripExtents(), Inkscape::UI::PathManipulator::_createGeometryFromControlPoints(), Inkscape::Extension::Internal::CairoRenderContext::_createHatchPainter(), Inkscape::UI::PathManipulator::_externalChange(), Inkscape::Extension::Internal::SvgBuilder::_getClip(), Inkscape::UI::Dialog::GuidelinePropertiesDialog::_modeChanged(), Inkscape::DrawingImage::_pickItem(), Inkscape::UI::Tools::EraserTool::_probeUnlinkCutClonedGroup(), Inkscape::DrawingText::_renderItem(), Inkscape::UI::Tools::TextTool::_setupText(), _ungroup_compensate_source_transform(), SPDesktop::DesktopAffine::_update(), Inkscape::UI::PathManipulator::_updateDragPoint(), Inkscape::LivePathEffect::LPETiling::addCanvasIndicators(), Inkscape::Extension::Internal::SvgBuilder::addShadedFill(), SPItem::adjust_paint_recursive(), Inkscape::UI::Dialog::CloneTiler::apply(), Inkscape::ObjectSet::applyAffine(), Inkscape::UI::Widget::Graphics::check_single_page(), Inkscape::CanvasItemBpath::closest_distance_to(), Inkscape::CanvasItemBpath::contains(), Geom::Parallelogram::contains(), Inkscape::DrawingMeshGradient::create_pattern(), SPImage::cropToArea(), Inkscape::UI::Tools::do_trace(), Inkscape::LivePathEffect::LPETiling::doAfterEffect(), Inkscape::LivePathEffect::LPEFillBetweenMany::doBeforeEffect(), Inkscape::LivePathEffect::LPEFillBetweenStrokes::doBeforeEffect(), Inkscape::LivePathEffect::LPEMeasureSegments::doBeforeEffect(), Inkscape::LivePathEffect::LPEMirrorSymmetry::doBeforeEffect(), Inkscape::LivePathEffect::LPEBool::doEffect(), Inkscape::LivePathEffect::LPETiling::doEffect_path(), Inkscape::LivePathEffect::LPEVonKoch::doEffect_path(), Inkscape::LivePathEffect::LPETiling::doEffect_path_post(), Inkscape::LivePathEffect::LPERoughHatches::doEffect_pwd2(), Inkscape::LivePathEffect::LPEFilletChamfer::doOnApply(), Inkscape::LivePathEffect::LPETiling::doOnVisibilityToggled(), PdfParser::doShadingPatternFillFallback(), SPItem::doWriteTransform(), Inkscape::UI::Tools::SpiralTool::drag(), Inkscape::UI::Tools::ArcTool::drag(), Inkscape::UI::Tools::RectTool::drag(), Inkscape::UI::Tools::StarTool::drag(), Inkscape::UI::TransformHandle::dragged(), SPItem::dt2i_affine(), SPShape::either_bbox(), Inkscape::UI::Widget::CairoGraphics::fast_snapshot_combine(), file_import(), Inkscape::LivePathEffect::findShadowedTime(), Inkscape::LivePathEffect::LPEPts2Ellipse::genFitEllipse(), SPBox3D::get_center_screen(), SPBox3D::get_corner_screen(), Inkscape::Filters::FilterUnits::get_matrix_display2pb(), Inkscape::Filters::FilterUnits::get_matrix_pb2display(), Inkscape::UI::Dialog::CloneTiler::get_transform(), Inkscape::Extension::Internal::getODFItemTransform(), i2i_affine(), Geom::intersect_polish_root(), Geom::Parallelogram::intersects(), PatternKnotHolderEntityScale::knot_grabbed(), KnotHolder::knot_grabbed_handler(), KnotHolder::knot_moved_handler(), HatchKnotHolderEntityScale::knot_set(), KnotHolder::knot_ungrabbed_handler(), Shape::MakeTweak(), SPUse::move_compensate(), GrDragger::moveMeshHandles(), PatternKnotHolderEntity::offset_to_cell(), Geom::Ellipse::operator*=(), Inkscape::LivePathEffect::GroupBBoxEffect::original_bbox(), Inkscape::UI::Widget::CairoGraphics::paint_widget(), Inkscape::LivePathEffect::PathParam::paste_param_path(), Inkscape::ObjectSet::pathBoolOp(), Inkscape::UI::PathManipulator::PathManipulator(), Inkscape::LivePathEffect::LPEMeasureSegments::processObjects(), Inkscape::ObjectSet::relink(), SPPath::removeTransformsRecursively(), Inkscape::Filters::FilterImage::render_cairo(), Inkscape::Filters::FilterTurbulence::render_cairo(), Inkscape::DrawingPattern::renderPattern(), SPConnEndPair::reroutePathFromLibavoid(), Inkscape::UI::Tools::ToolBase::root_handler(), SPItem::rotate_rel(), Inkscape::UI::Dialog::rotateAround(), SPGroup::scaleChildItemsRec(), SPGradient::set_gs2d_matrix(), SPItem::set_i2d_affine(), Inkscape::UI::Tools::CalligraphicTool::set_to_accumulated(), SPImage::set_transform(), SPText::set_transform(), SPGenericEllipse::set_transform(), SPFlowtext::set_transform(), SPRect::set_transform(), SPSpiral::set_transform(), SPStar::set_transform(), Inkscape::UI::PathManipulator::setControlsTransform(), Inkscape::UI::Tools::MeasureTool::setLine(), Inkscape::ObjectSet::setMask(), Inkscape::UI::Tools::MeasureTool::setMeasureItem(), Inkscape::UI::Tools::MeasureTool::setPoint(), Inkscape::UI::Widget::RegisteredTransformedPoint::setValue(), Inkscape::BooleanBuilder::shape_commit(), Inkscape::Text::Layout::showGlyphs(), Inkscape::UI::Tools::MeasureTool::showInfoBox(), SPItem::skew_rel(), Inkscape::UI::Widget::CairoGraphics::snapshot_combine(), Inkscape::UI::Widget::Stores::snapshot_combine(), Inkscape::Extension::Internal::sp_asbitmap_render(), Inkscape::UI::Tools::sp_flood_do_flood_fill(), sp_gradient_convert_to_userspace(), sp_gradient_pattern_common_setup(), sp_import_document(), sp_item_gradient_set_coords(), sp_offset_move_compensate(), sp_selection_paste_impl(), Inkscape::UI::Tools::sp_spray_transform_path(), sp_te_get_position_by_coords(), sp_usepath_move_compensate(), Inkscape::UI::Tools::spdc_create_single_dot(), Inkscape::UI::Tools::spdc_flush_white(), Inkscape::LivePathEffect::LPESlice::splititem(), Inkscape::SelTrans::stamp(), Inkscape::UI::Widget::Stores::take_snapshot(), Geom::TEST(), Inkscape::ObjectSet::tile(), Inkscape::LivePathEffect::LPEBendPath::transform_multiply(), Inkscape::LivePathEffect::LPEBool::transform_multiply(), Inkscape::LivePathEffect::LPEOffset::transform_multiply(), Inkscape::LivePathEffect::LPEFillBetweenMany::transform_multiply_nested(), Inkscape::LivePathEffect::LPEFillBetweenStrokes::transform_multiply_nested(), Inkscape::UI::Tools::tweak_colors_in_gradient(), Inkscape::DrawingItem::update(), Inkscape::LivePathEffect::LPEFilletChamfer::updateAmount(), and wrap_transforms().

isHShear()

bool Geom::Affine::isHShear

(

Coord

eps = EPSILON

)

const

Check whether this matrix represents pure horizontal shearing.

Parameters

eps	Numerical tolerance

Returns

True iff the matrix is of the form \(\left[\begin{array}{ccc} 1 & 0 & 0 \\ k & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\).

Definition at line 256 of file affine.cpp.

References _c, and are_near().

Referenced by sp_svg_transform_write(), and Geom::TEST().

isIdentity()

bool Geom::Affine::isIdentity

(

Coord

eps = EPSILON

)

const

Check whether this matrix is an identity matrix.

Parameters

eps	Numerical tolerance

Returns

True iff the matrix is of the form \(\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

Definition at line 109 of file affine.cpp.

References _c, and are_near().

Referenced by Inkscape::ObjectSet::applyAffine(), Inkscape::UI::Tools::do_trace(), SPItem::doWriteTransform(), SPMeshNodeArray::fill_box(), Inkscape::Extension::Internal::formatTransform(), Inkscape::Filters::FilterSlot::get_result(), SPFlowtext::getAsText(), SPItem::invoke_print(), Inkscape::DrawingCache::prepare(), refresh_offset_source(), SPGroup::scaleChildItemsRec(), Inkscape::DrawingGroup::setChildTransform(), Inkscape::DrawingPattern::setPatternToUserTransform(), Inkscape::DrawingItem::setTransform(), sp_item_group_ungroup(), sp_selection_paste_impl(), sp_svg_transform_write(), Geom::TEST(), Inkscape::LivePathEffect::LPEFillBetweenMany::transform_multiply_nested(), Inkscape::LivePathEffect::LPEFillBetweenStrokes::transform_multiply_nested(), and write_transform().

isNonzeroHShear()

bool Geom::Affine::isNonzeroHShear

(

Coord

eps = EPSILON

)

const

Check whether this matrix represents pure, nonzero horizontal shearing.

Parameters

eps	Numerical tolerance

Returns

True iff the matrix is of the form \(\left[\begin{array}{ccc} 1 & 0 & 0 \\ k & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\) and \(k \neq 0\).

Definition at line 268 of file affine.cpp.

References _c, and are_near().

Referenced by Geom::TEST().

isNonzeroNonpureRotation()

bool Geom::Affine::isNonzeroNonpureRotation

(

Coord

eps = EPSILON

)

const

Check whether this matrix represents a non-zero rotation about any point.

Parameters

eps	Numerical tolerance

Returns

True iff the matrix is of the form \(\left[\begin{array}{ccc} a & b & 0 \\ -b & a & 0 \\ c & d & 1 \end{array}\right]\), \(a^2 + b^2 = 1\) and \(a \neq 1\).

Definition at line 233 of file affine.cpp.

References _c, and are_near().

Referenced by Geom::TEST().

isNonzeroRotation()

bool Geom::Affine::isNonzeroRotation

(

Coord

eps = EPSILON

)

const

Check whether this matrix represents a pure, nonzero rotation.

Parameters

eps	Numerical tolerance

Returns

True iff the matrix is of the form \(\left[\begin{array}{ccc} a & b & 0 \\ -b & a & 0 \\ 0 & 0 & 1 \end{array}\right]\), \(a^2 + b^2 = 1\) and \(a \neq 1\).

Definition at line 219 of file affine.cpp.

References _c, and are_near().

Referenced by Geom::TEST().

isNonzeroScale()

bool Geom::Affine::isNonzeroScale

(

Coord

eps = EPSILON

)

const

Check whether this matrix represents pure, nonzero scaling.

Parameters

eps	Numerical tolerance

Returns

True iff the matrix is of the form \(\left[\begin{array}{ccc} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & 1 \end{array}\right]\) and \(a, b \neq 1\).

Definition at line 160 of file affine.cpp.

References _c, are_near(), and isSingular().

Referenced by Geom::TEST().

isNonzeroTranslation()

bool Geom::Affine::isNonzeroTranslation

(

Coord

eps = EPSILON

)

const

Check whether this matrix represents a pure nonzero translation.

Parameters

eps	Numerical tolerance

Returns

True iff the matrix is of the form \(\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & 1 \end{array}\right]\) and \(a, b \neq 0\)

Definition at line 134 of file affine.cpp.

References _c, and are_near().

Referenced by Geom::TEST().

isNonzeroUniformScale()

bool Geom::Affine::isNonzeroUniformScale

(

Coord

eps = EPSILON

)

const

Check whether this matrix represents pure, nonzero uniform scaling.

Parameters

eps	Numerical tolerance

Returns

True iff the matrix is of the form \(\left[\begin{array}{ccc} a_1 & 0 & 0 \\ 0 & a_2 & 0 \\ 0 & 0 & 1 \end{array}\right]\) where \(|a_1| = |a_2|\) and \(a_1, a_2 \neq 1\).

Definition at line 189 of file affine.cpp.

References _c, are_near(), and isSingular().

Referenced by SPFlowtext::set_transform(), and Geom::TEST().

isNonzeroVShear()

bool Geom::Affine::isNonzeroVShear

(

Coord

eps = EPSILON

)

const

Check whether this matrix represents pure, nonzero vertical shearing.

Parameters

eps	Numerical tolerance

Returns

True iff the matrix is of the form \(\left[\begin{array}{ccc} 1 & k & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\) and \(k \neq 0\).

Definition at line 294 of file affine.cpp.

References _c, and are_near().

Referenced by Geom::TEST().

isRotation()

bool Geom::Affine::isRotation

(

Coord

eps = EPSILON

)

const

Check whether this matrix represents a pure rotation.

Parameters

eps	Numerical tolerance

Returns

True iff the matrix is of the form \(\left[\begin{array}{ccc} a & b & 0 \\ -b & a & 0 \\ 0 & 0 & 1 \end{array}\right]\) and \(a^2 + b^2 = 1\).

Definition at line 206 of file affine.cpp.

References _c, and are_near().

Referenced by sp_svg_transform_write(), Geom::TEST(), and wrap_transforms().

isScale()

bool Geom::Affine::isScale

(

Coord

eps = EPSILON

)

const

Check whether this matrix represents pure scaling.

Parameters

eps	Numerical tolerance

Returns

True iff the matrix is of the form \(\left[\begin{array}{ccc} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & 1 \end{array}\right]\).

Definition at line 147 of file affine.cpp.

References _c, are_near(), and isSingular().

Referenced by Geom::Ellipse::operator*=(), sp_svg_transform_write(), Geom::TEST(), and wrap_transforms().

isSingular()

bool Geom::Affine::isSingular

(

Coord

eps = EPSILON

)

const

Check whether this matrix is singular.

Singular matrices have no inverse, which means that applying them to a set of points results in a loss of information.

Parameters

eps	Numerical tolerance

Returns

True iff the determinant is near zero.

Definition at line 377 of file affine.cpp.

References are_near(), and det().

Referenced by Inkscape::UI::ControlPointSelection::_pointDragged(), Inkscape::UI::Dialog::Transformation::applyPageTransform(), Geom::Parallelogram::contains(), Inkscape::UI::TransformHandle::dragged(), Geom::Parallelogram::intersects(), isNonzeroScale(), isNonzeroUniformScale(), isScale(), isUniformScale(), isZoom(), preservesAngles(), Inkscape::DrawingItem::render(), Inkscape::UI::Widget::RegisteredTransformedPoint::setTransform(), and Geom::TEST().

isTranslation()

bool Geom::Affine::isTranslation

(

Coord

eps = EPSILON

)

const

Check whether this matrix represents a pure translation.

Will return true for the identity matrix, which represents a zero translation.

Parameters

eps	Numerical tolerance

Returns

True iff the matrix is of the form \(\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & 1 \end{array}\right]\)

Definition at line 123 of file affine.cpp.

References _c, and are_near().

Referenced by Inkscape::Filters::FilterSlot::_get_transformed_source_graphic(), Inkscape::ObjectSet::applyAffine(), SPItem::doWriteTransform(), Inkscape::Filters::FilterSlot::FilterSlot(), SPPage::movePage(), Inkscape::DrawingCache::prepare(), sp_svg_transform_write(), Geom::TEST(), Inkscape::ObjectSet::toSymbol(), Inkscape::LivePathEffect::LPEOffset::transform_multiply(), and wrap_transforms().

isUniformScale()

bool Geom::Affine::isUniformScale

(

Coord

eps = EPSILON

)

const

Check whether this matrix represents pure uniform scaling.

Parameters

eps	Numerical tolerance

Returns

True iff the matrix is of the form \(\left[\begin{array}{ccc} a_1 & 0 & 0 \\ 0 & a_2 & 0 \\ 0 & 0 & 1 \end{array}\right]\) where \(|a_1| = |a_2|\).

Definition at line 174 of file affine.cpp.

References _c, are_near(), and isSingular().

Referenced by SPLPEItem::optimizeTransforms(), SPSpiral::set_transform(), SPStar::set_transform(), Geom::TEST(), and wrap_transforms().

isVShear()

bool Geom::Affine::isVShear

(

Coord

eps = EPSILON

)

const

Check whether this matrix represents pure vertical shearing.

Parameters

eps	Numerical tolerance

Returns

True iff the matrix is of the form \(\left[\begin{array}{ccc} 1 & k & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\).

Definition at line 281 of file affine.cpp.

References _c, and are_near().

Referenced by sp_svg_transform_write(), and Geom::TEST().

isZoom()

bool Geom::Affine::isZoom

(

Coord

eps = EPSILON

)

const

Check whether this matrix represents zooming.

Zooming is any combination of translation and uniform non-flipping scaling. It preserves angles, ratios of distances between arbitrary points and unit vectors of line segments.

Parameters

eps	Numerical tolerance

Returns

True iff the matrix is invertible and of the form \(\left[\begin{array}{ccc} a & 0 & 0 \\ 0 & a & 0 \\ b & c & 1 \end{array}\right]\).

Definition at line 310 of file affine.cpp.

References _c, are_near(), and isSingular().

Referenced by Geom::TEST().

operator*=() [1/7]

Affine & Geom::Affine::operator*=

(

Affine const &

o

)

Combine this transformation with another one.

After this operation, the matrix will correspond to the transformation obtained by first applying the original version of this matrix, and then applying m.

Definition at line 442 of file affine.cpp.

References _c.

operator*=() [2/7]

Affine & Geom::Affine::operator*=

(

HShear const &

h

)

Combine this transformation with horizontal shearing (skew).

Definition at line 119 of file transforms.cpp.

References _c, and Geom::ShearBase< S >::f.

operator*=() [3/7]

Affine & Geom::Affine::operator*=

(

Rotate const &

r

)

Combine this transformation a rotation.

Definition at line 111 of file transforms.cpp.

References Affine().

operator*=() [4/7]

Affine & Geom::Affine::operator*=

(

Scale const &

s

)

Combine this transformation with scaling.

Definition at line 103 of file transforms.cpp.

References _c, Geom::X, and Geom::Y.

operator*=() [5/7]

Affine & Geom::Affine::operator*=

(

Translate const &

t

)

Combine this transformation with a translation.

Definition at line 96 of file transforms.cpp.

References _c, Geom::X, and Geom::Y.

operator*=() [6/7]

Affine & Geom::Affine::operator*=

(

VShear const &

v

)

Combine this transformation with vertical shearing (skew).

Definition at line 127 of file transforms.cpp.

References _c, and Geom::SL::v.

operator*=() [7/7]

Affine & Geom::Affine::operator*=

(

Zoom const &

z

)

Definition at line 134 of file transforms.cpp.

References _c, Geom::Zoom::_scale, Geom::Zoom::_trans, Geom::X, and Geom::Y.

operator==()

bool Geom::Affine::operator== ( Affine const &  o ) const

inline

Definition at line 109 of file affine.h.

References _c.

operator[]() [1/2]

Coord & Geom::Affine::operator[] ( unsigned  i )

inline

Definition at line 95 of file affine.h.

References _c.

operator[]() [2/2]

Coord Geom::Affine::operator[] ( unsigned  i ) const

inline

Access a coefficient by its index.

Definition at line 94 of file affine.h.

References _c.

preservesAngles()

bool Geom::Affine::preservesAngles

(

Coord

eps = EPSILON

)

const

Check whether the transformation preserves angles between lines.

This means that the transformation can be any combination of translation, uniform scaling, rotation and flipping.

Parameters

eps	Numerical tolerance

Returns

True iff the matrix is of the form \(\left[\begin{array}{ccc} a & b & 0 \\ -b & a & 0 \\ c & d & 1 \end{array}\right]\) or \(\left[\begin{array}{ccc} -a & b & 0 \\ b & a & 0 \\ c & d & 1 \end{array}\right]\).

Definition at line 339 of file affine.cpp.

References _c, are_near(), and isSingular().

Referenced by Inkscape::LivePathEffect::LPEMeasureSegments::doBeforeEffect(), and Geom::TEST().

preservesArea()

bool Geom::Affine::preservesArea

(

Coord

eps = EPSILON

)

const

Check whether the transformation preserves areas of polygons.

This means that the transformation can be any combination of translation, rotation, shearing and squeezing (non-uniform scaling such that the absolute value of the product of Y-scale and X-scale is 1).

Parameters

eps	Numerical tolerance

Returns

True iff \(|\det A| = 1\).

Definition at line 321 of file affine.cpp.

References are_near(), and descrim2().

Referenced by Geom::TEST().

preservesDistances()

bool Geom::Affine::preservesDistances

(

Coord

eps = EPSILON

)

const

Check whether the transformation preserves distances between points.

This means that the transformation can be any combination of translation, rotation and flipping.

Parameters

eps	Numerical tolerance

Returns

True iff the matrix is of the form \(\left[\begin{array}{ccc} a & b & 0 \\ -b & a & 0 \\ c & d & 1 \end{array}\right]\) or \(\left[\begin{array}{ccc} -a & b & 0 \\ b & a & 0 \\ c & d & 1 \end{array}\right]\) and \(a^2 + b^2 = 1\).

Definition at line 359 of file affine.cpp.

References _c, and are_near().

Referenced by Geom::TEST().

rotationCenter()

Point Geom::Affine::rotationCenter

(

)

const

For a (possibly non-pure) non-zero-rotation matrix, calculate the rotation center.

Precondition

The matrix must be a non-zero-rotation matrix to prevent division by zero, see isNonzeroNonpureRotation().

Returns

The rotation center x, the solution to the equation \(A x = x\).

Definition at line 243 of file affine.cpp.

References _c.

Referenced by Geom::TEST().

setExpansionX()

void Geom::Affine::setExpansionX

(

Coord

val

)

Definition at line 75 of file affine.cpp.

References _c, and expansionX().

Referenced by Inkscape::UI::Tools::spdc_check_for_and_apply_waiting_LPE().

setExpansionY()

void Geom::Affine::setExpansionY

(

Coord

val

)

Definition at line 85 of file affine.cpp.

References _c, and expansionY().

Referenced by Inkscape::UI::Tools::spdc_check_for_and_apply_waiting_LPE().

setIdentity()

void Geom::Affine::setIdentity

(

)

Sets this matrix to be the Identity Affine.

Definition at line 96 of file affine.cpp.

References _c.

Referenced by Inkscape::UI::ControlPointSelection::_pointDragged(), Inkscape::UI::Dialog::PolarArrangeTab::arrange(), SPMeshNodeArray::fill_box(), Inkscape::UI::TransformHandle::grabbed(), inverse(), SPUse::move_compensate(), Inkscape::DrawingCache::prepare(), Inkscape::SelTrans::SelTrans(), Inkscape::Text::Layout::showGlyphs(), sp_offset_move_compensate(), SPPattern::update_view(), and wrap_transforms().

setTranslation()

void Geom::Affine::setTranslation

(

Point const &

loc

)

Sets the translation imparted by the Affine.

Definition at line 56 of file affine.cpp.

References _c.

Referenced by Inkscape::Extension::Internal::SvgBuilder::_flushText(), Inkscape::LivePathEffect::LPECurveStitch::doEffect_path(), Inkscape::UI::Widget::PatternEditor::get_selected_transform(), Inkscape::UI::Tools::spdc_check_for_and_apply_waiting_LPE(), Geom::Ellipse::unitCircleTransform(), Inkscape::DrawingItem::update(), withoutTranslation(), and wrap_transforms().

setXAxis()

void Geom::Affine::setXAxis

(

Point const &

vec

)

Definition at line 45 of file affine.cpp.

References _c.

Referenced by Inkscape::LivePathEffect::LPECurveStitch::doEffect_path(), and wrap_transforms().

setYAxis()

void Geom::Affine::setYAxis

(

Point const &

vec

)

Definition at line 50 of file affine.cpp.

References _c.

Referenced by Inkscape::LivePathEffect::LPECurveStitch::doEffect_path(), and wrap_transforms().

translation()

Point Geom::Affine::translation

(

)

const

Gets the translation imparted by the Affine.

Definition at line 41 of file affine.cpp.

References _c.

Referenced by SPMarker::get_marker_transform(), Inkscape::DrawingCache::prepare(), Inkscape::Text::Layout::show(), and wrap_transforms().

withoutTranslation()

Affine Geom::Affine::withoutTranslation ( ) const

inline

Definition at line 169 of file affine.h.

References setTranslation().

Referenced by Inkscape::CanvasItemGuideLine::_render(), Inkscape::DrawingText::_renderItem(), Inkscape::CanvasItemGridXY::_update(), Inkscape::CanvasItemGridTiles::_update(), Inkscape::DrawingRadialGradient::create_pattern(), Inkscape::LivePathEffect::LPETiling::doAfterEffect(), Inkscape::LivePathEffect::LPECurveStitch::doEffect_path(), Inkscape::UI::Tools::MarkerTool::get_marker_transform(), Inkscape::Extension::Internal::PrintWmf::image(), PatternKnotHolderEntityScale::knot_grabbed(), Inkscape::LivePathEffect::VectorParam::param_transform_multiply(), SPFlowtext::set_transform(), SPSpiral::set_transform(), SPStar::set_transform(), SPUse::snappoints(), Inkscape::Extension::Internal::LaTeXTextRenderer::sp_flowtext_render(), Inkscape::UI::Tools::sp_spray_recursive(), sp_svg_transform_write(), Inkscape::Extension::Internal::LaTeXTextRenderer::sp_text_render(), Inkscape::UI::Tools::sp_tweak_dilate_recursive(), Geom::TEST(), Geom::Ellipse::unitTangentAt(), and vectorStretch().

xAxis()

Point Geom::Affine::xAxis

(

)

const

Definition at line 32 of file affine.cpp.

References _c.

Referenced by Geom::Parallelogram::isSheared(), Inkscape::Extension::Internal::LaTeXTextRenderer::sp_flowtext_render(), Inkscape::Extension::Internal::LaTeXTextRenderer::sp_text_render(), Inkscape::UI::Widget::PatternEditor::update_widgets_from_pattern(), and wrap_transforms().

yAxis()

Point Geom::Affine::yAxis

(

)

const

Definition at line 36 of file affine.cpp.

References _c.

Referenced by Geom::Parallelogram::isSheared(), Inkscape::UI::Widget::PatternEditor::update_widgets_from_pattern(), and wrap_transforms().

Friends And Related Function Documentation

are_near()

bool are_near ( Affine const &  a, Affine const &  b, Coord  eps  )

related

Nearness predicate for affine transforms.

Returns

True if all entries of matrices are within eps of each other.

Definition at line 504 of file affine.cpp.

References Geom::are_near().

Referenced by isHShear(), isIdentity(), isNonzeroHShear(), isNonzeroNonpureRotation(), isNonzeroRotation(), isNonzeroScale(), isNonzeroTranslation(), isNonzeroUniformScale(), isNonzeroVShear(), isRotation(), isScale(), isSingular(), isTranslation(), isUniformScale(), isVShear(), isZoom(), preservesAngles(), preservesArea(), and preservesDistances().

elliptic_quadratic_form()

Affine elliptic_quadratic_form ( Affine const &  m )

related

Given a matrix m such that unit_circle = m*x, this returns the quadratic form x*A*x = 1.

Definition at line 461 of file affine.cpp.

References Hsluv::m.

identity()

Affine identity ( )

related

Create an identity matrix.

Returns

The matrix \(\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\).

Definition at line 222 of file affine.h.

operator<<()

std::ostream & operator<< ( std::ostream &  out_file, const Geom::Affine &  m  )

related

Print out the Affine (for debugging).

Definition at line 188 of file affine.h.

References Hsluv::m.

Member Data Documentation

_c

Coord Geom::Affine::_c[6]

private

Definition at line 71 of file affine.h.

Referenced by Affine(), det(), expansionX(), expansionY(), inverse(), isHShear(), isIdentity(), isNonzeroHShear(), isNonzeroNonpureRotation(), isNonzeroRotation(), isNonzeroScale(), isNonzeroTranslation(), isNonzeroUniformScale(), isNonzeroVShear(), isRotation(), isScale(), isTranslation(), isUniformScale(), isVShear(), isZoom(), operator*=(), operator==(), operator[](), preservesAngles(), preservesDistances(), rotationCenter(), setExpansionX(), setExpansionY(), setIdentity(), setTranslation(), setXAxis(), setYAxis(), translation(), xAxis(), and yAxis().

---

The documentation for this class was generated from the following files:

• /builds/inkscape/inkscape/src/3rdparty/2geom/include/2geom/affine.h
• /builds/inkscape/inkscape/src/3rdparty/2geom/src/2geom/affine.cpp
• /builds/inkscape/inkscape/src/3rdparty/2geom/src/2geom/transforms.cpp